Suppose that $X_{1}$ and $X_{2}$ denote the output of rolls of two independent dices that can each take integer values $\{1,2,3,4,5,6\}$ with probability $1 / 6$ for each outcome. Further, $U$ denotes a continuous random variable that is independent of $X_{1}$ and $X_{2}$ and is uniformly distributed in the interval $[0,1]$.
Suppose that the sum of the three random variables, that is, $X_{1}+X_{2}+U,$ equals $6.63.$ Conditioned on this sum what is the probability that $X_{1}$ equals $2?$
- $2.21$
- $3$
- $1 / 6$
- $1 / 5$
- $1 / 3$